| L(s) = 1 | − 8·11-s − 32·71-s + 30·81-s − 108·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + ⋯ |
| L(s) = 1 | − 2.41·11-s − 3.79·71-s + 10/3·81-s − 9.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + 0.0646·239-s + 0.0644·241-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6261408814\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6261408814\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 4 p T^{4} + 646 T^{8} - 4 p^{5} T^{12} + p^{8} T^{16} \) |
| good | 3 | \( ( 1 - 5 p T^{4} + 112 T^{8} - 5 p^{5} T^{12} + p^{8} T^{16} )^{2} \) |
| 11 | \( ( 1 + T + 18 T^{2} + p T^{3} + p^{2} T^{4} )^{8} \) |
| 13 | \( ( 1 - 19 p T^{4} + 68800 T^{8} - 19 p^{5} T^{12} + p^{8} T^{16} )^{2} \) |
| 17 | \( ( 1 - 135 T^{4} + 162992 T^{8} - 135 p^{4} T^{12} + p^{8} T^{16} )^{2} \) |
| 19 | \( ( 1 - 4 T^{2} - 362 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 23 | \( ( 1 - 4 p T^{4} - 417402 T^{8} - 4 p^{5} T^{12} + p^{8} T^{16} )^{2} \) |
| 29 | \( ( 1 - 39 T^{2} + 2024 T^{4} - 39 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 31 | \( ( 1 - 12 T^{2} + 870 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 37 | \( ( 1 + 626 T^{4} + p^{4} T^{8} )^{4} \) |
| 41 | \( ( 1 - 84 T^{2} + 4038 T^{4} - 84 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 43 | \( ( 1 - 124 T^{4} - 6323354 T^{8} - 124 p^{4} T^{12} + p^{8} T^{16} )^{2} \) |
| 47 | \( ( 1 + 7553 T^{4} + 23882080 T^{8} + 7553 p^{4} T^{12} + p^{8} T^{16} )^{2} \) |
| 53 | \( ( 1 - 6748 T^{4} + 22133926 T^{8} - 6748 p^{4} T^{12} + p^{8} T^{16} )^{2} \) |
| 59 | \( ( 1 + 76 T^{2} + 4054 T^{4} + 76 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 61 | \( ( 1 - 132 T^{2} + 10710 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 67 | \( ( 1 + 9092 T^{4} + 57308326 T^{8} + 9092 p^{4} T^{12} + p^{8} T^{16} )^{2} \) |
| 71 | \( ( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{8} \) |
| 73 | \( ( 1 - 2940 T^{4} + 23706182 T^{8} - 2940 p^{4} T^{12} + p^{8} T^{16} )^{2} \) |
| 79 | \( ( 1 - 99 T^{2} + 13976 T^{4} - 99 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 83 | \( ( 1 - 17020 T^{4} + 141533734 T^{8} - 17020 p^{4} T^{12} + p^{8} T^{16} )^{2} \) |
| 89 | \( ( 1 + 244 T^{2} + 29638 T^{4} + 244 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 97 | \( ( 1 + 5577 T^{4} + 169216016 T^{8} + 5577 p^{4} T^{12} + p^{8} T^{16} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.44277665838068231919145665501, −2.32545436842459283160918416845, −2.21574912088882408240270482758, −2.20947869122043575601815065610, −2.16098837889267984349405378872, −2.09712095013169090183560635206, −1.97059069266628310960631849752, −1.95806496177385972264585853452, −1.89850315216386053525017881185, −1.81009038058949849473405599635, −1.60142988333774182713016272575, −1.52525088158962568921434034412, −1.39824268750153842089570474364, −1.25651760914422651040323370888, −1.23787827953158241306384569773, −1.18688640597848024160435624178, −1.13910070527970706812104428059, −1.10502328347376794260519797114, −1.05509267414508224379523946744, −0.792433260004610187260843592829, −0.49792957138180852258996187267, −0.46691027141751224710254300592, −0.29011712315015713448232922683, −0.13893647899326867015813084062, −0.12644937876407215940539991924,
0.12644937876407215940539991924, 0.13893647899326867015813084062, 0.29011712315015713448232922683, 0.46691027141751224710254300592, 0.49792957138180852258996187267, 0.792433260004610187260843592829, 1.05509267414508224379523946744, 1.10502328347376794260519797114, 1.13910070527970706812104428059, 1.18688640597848024160435624178, 1.23787827953158241306384569773, 1.25651760914422651040323370888, 1.39824268750153842089570474364, 1.52525088158962568921434034412, 1.60142988333774182713016272575, 1.81009038058949849473405599635, 1.89850315216386053525017881185, 1.95806496177385972264585853452, 1.97059069266628310960631849752, 2.09712095013169090183560635206, 2.16098837889267984349405378872, 2.20947869122043575601815065610, 2.21574912088882408240270482758, 2.32545436842459283160918416845, 2.44277665838068231919145665501
Plot not available for L-functions of degree greater than 10.
